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Development of an Improved Prediction Model for
Indoor Rolling Noise
Matthew Edwards, Fabien Chevillotte, François-Xavier Bécot, Luc Jaouen,
Nicolas Totaro
To cite this version:
DEVELOPMENT OF AN IMPROVED PREDICTION MODEL FOR
INDOOR ROLLING NOISE
M. Edwards
1F. Chevillotte
1F.-X. B´ecot
1L. Jaouen
1N. Totaro
21
Matelys Research Lab, 7 Rue des Maraˆıchers, Bˆat B, 69120 Vaulx-en-Velin , France
2INSA—Lyon 20 Avenue Albert Einstein, 69621 Villeurbanne cedex, France
matthew.edwards@matelys.com
ABSTRACT
This work presents a prediction model for rolling noise in multi-story buildings, such as that generated by a rolling delivery trolley. Until now, mechanical excitation in multi-story buildings has been limited to impact sources such as the tapping machine. Rolling noise models have been lim-ited to outdoor sources such as trains and automotive vehi-cles. The model presented here is able to predict the nor-malized sound pressure level in the reception room below generated by a trolley rolling in the emission room above, as well as the relative sound benefit of adding a covering to the floor upon which the trolley is rolling. The model can also represent the physical phenomena unique to in-door rolling noise, taking into account influencing factors such as the roughness of the wheel and the floor, the ma-terial and geometric properties of the wheel and the floor, the rolling velocity of the trolley, and the load on the trol-ley. The model may be used as a tool to investigate how different flooring systems (including multi-layer systems) respond to rolling excitation, for the purpose of developing multi-story building solutions which are better equipped to combat this kind of noise source.
1. INTRODUCTION
Structure-borne noise insulation is a field of building acoustics which has seen a large amount of research in recent decades. By far and away, the most common method of evaluating a building’s performance in mitigat-ing structure-borne noise is linked to impact noise. This is done primarily through the use of a tapping machine [1], though other methods such as the rubber ball [2] (see ap-pendix F), walking noise [3], and even rainfall noise [2] are sometimes used. Recommendations for prediction of upward and lateral radiation from airborne and structure-borne excitation in heavy-weight buildings exists as well [4]. No standard currently exists for transmission to ad-jacent rooms for light-weight buildings. However, other sources exist beyond these, such as rolling noise.
Indoor rolling noise can take many forms: rolling office chairs, children’s toys, and suitcases being a few examples of such structure-borne excitation sources. The most an-noying, however, are possibly rolling trolleys or delivery carts, an example of which is shown in Figure 1.
Figure 1. A typical rolling delivery trolley.
Figure 2 shows the acoustic excitation due to a typical tapping machine and a typical rolling trolley on a classical concrete floor, in addition to the attenuation of a classical floating floor. The spectrum of impact noise is quite dif-ferent than that of rolling noise, having the majority of its acoustic energy in the low frequency range. Furthermore, the attenuation of a classical floating floor is inverse to said rolling noise profile. As such, it does not provide sufficient insulation on its own without additional treatment.
Figure 2. Comparison of the spectra of tapping noise and rolling noise of a classical concrete floor, as well as the attenuation of a classical floating floor (140 mm concrete slab + a decoupling layer + 40 mm screed) [5].
dif-ferent than that of impact noise, the solutions developed to help against impact noise can be sometimes ineffective against rolling noise. Additionally, impact noise model-ing techniques cannot accurately predict the level of rollmodel-ing noise generated in the same environment. For this reason, a unique method of modeling and predicting indoor rolling noise is needed.
Rolling noise has indeed been thoroughly investigated for decades, but uniquely in the context of vehicle tire/road and train wheel/rail contact. Models which aim at predict-ing these types of noise exist (see for example [6–8] for tire and [9–12] for train), but they cannot be easily adapted to work for indoor rolling noise due to differences in the un-derlying phenomena which cause such noise (mainly the propagation of the sound to the surrounding environment, which is quite different between roads, trains tracks and indoor buildings).
Previously, we developed a simple model for predict-ing rollpredict-ing noise in an indoor environment [5, 13]. This model calculates the contact force between the wheel and the floor in the time domain, and uses it as the injected force into the floor to predict the resulting radiated sound power. Here we present an improved version of this model, which greatly expands in scope and application the capa-bilities of predicting indoor rolling noise. The improved model is capable of accounting for floors of various types and layered constructions, as well as discrete irregularities (beyond small-scale surface roughness) such as wheel flats and floor joints. In the sections that follow, the model is presented, along with a comparison with experimental test results.
2. MODEL DEVELOPMENT
Figure 3 shows the general outline of the rolling noise model. Operating in the time domain and in three spa-tial dimensions, the model uses the influencing properties of the wheel and floor to calculate the contact force be-tween the two bodies for each discretized moment in time throughout the rolling event. These include the wheel and floor roughness profiles, the wheel and floor material prop-erties (Young’s modulus and Poisson’s ratio), the speed of the trolley, the mass of the trolley (plus added load), and geometric profiles of the wheel and floor. Such profiles in-clude not only the dimensions of the wheel, but also any discrete irregularities such as wheel flats or floor joints.
The contact force at each discretized moment in time is used as input into the dynamic model, where the re-sulting movement of the wheel and floor are calculated in response to the excitation injected force. This influences what the exact roughness profile will be for the follow-ing time step, as the wheel and floor continuously move in relation to one another, influencing how much (or how little) contact exists between them. Once the contact force has been calculated for each time step in the rolling event, the model is converted to the frequency domain, where the blocked force spectra is used in conjunction with the wheel and floor impedances to convert the blocked force to an in-jected force. From there the resulting radiated sound power
may be calculated in the reception room through use of the transfer matrix method.
Figure 3. General outline of the rolling noise model.
2.1 Dynamic Model
The dynamic model, shown in Figure 4, is based on a wheel modeled as a spring-mass damper system resting on a rigid floor. The mass M is the mass of the trolley plus added load, divided by the total number of wheels on the trolley. The stiffness K is the equivalent elastic stiffness of the wheel, which is a function of the wheel’s geome-try and elastic properties. A presentation of how such a stiffness is calculated is given in [14]. A small amount of viscous damping C is added to be able to tune the model and avoid instabilities. The wheel is represented by its re-ceptance GW(f ) and Green’s function gW(t), which are a
measure of the wheel’s vertical displacement in response to an impulse force input.
GW(f ) =
1
−(2πf )2M + i(2πf )C + K (1)
gW(t) =F−1(GW(f )) (2)
whereF−1represents the inverse Fourier transform.
2.2 Contact Model
Figure 4. Dynamic model of the wheel/floor contact. following equations u0R(x, x0, y0) = u0− zW(x) − ς(x, x0, y0)− ξW(x, x0, y0) + ξF(x, x0) (3) ς(x, x0, y0) = ςF(x, x0, y0) − ςW(x, x0, y0) (4) σR(x, x0, y0) = ( σ0 qu0 R(x,x0,y0) u0 u 0 R(x, x0, y0) > 0 0 u0R(x, x0, y0) ≤ 0 (5) FR(x) = Z Z σR(x, x0, y0)dx0dy0 (6) zW = gW(1)(FR(x) − Q) + gW ∗ (FR− Q) (7) x = vt (8) A bed of independent, locally reacting springs is placed between the wheel and the floor, and their deflection used to estimate the stress profile σRin the area of contact for
each time step t. The time step is converted to a position step by use of the trolley speed v.
Figure 5. Contact model used for estimating the contact stresses between the wheel and the floor
The geometry of this interaction is determined by the total interpenetration profile u0Rbetween the floor and the wheel at a given wheel center position x, which is made
up of the static Hertzian interpenetration in the absence of roughness (constant for all x), the wheel profile ξW, floor
profile ξF, and relative roughness profile ς. The relative
roughness profile is a difference of the individual wheel and floor roughness profiles at a given x.
Harris provides a thorough explanation of the static Hertzian deflection u0in [18] (see chapter 6). For a
cylin-drical wheel rolling on a flat floor, the deflection is u0=
4Q
E0πw (9)
where Q is the effective mass M expressed as a load in Newtons, w is the wheel width (in the y direction), and E0 is the apparent Young’s modulus between the wheel and the floor, defined by
E0= 1 − ν 2 W EW +1 − ν 2 F EF −1 (10) where E and ν are the Young’s modulus and Poisson’s ra-tio of the wheel and floor (denoted by the appropriate sub-scripts).
Typically, the wheel profile will be simply defined by it’s radii of curvature,
ξW(x, x0, y0) = x02 2r0 x + y 02 2r0 y (11) The floor profile will typically be zero for all x. However, in the presence of discrete irregularities such as wheel flat spots or floor joints, they may differ.
In order to ensure agreement with Hertzian contact me-chanics, the wheel radii (both longitudinal rx and
trans-verse ry) need to be reduced by unique factors which
de-pend on the wheel’s geometry, yielding reduced radii rx0 and r0y. A full explanation is again given in chapter 6
in [18]. For a cylindrical wheel, the reduction is one half the original wheel radius. This modified radius must be used when defining the wheel profile ξW.
For a given moment in time, the wheel center is at a position x in the global coordinate system. The local co-ordinate system is positioned such that x0(0) = x, thus the origin of the local system follows the wheel center in the x dimension. For a given x, the roughness profile ς, wheel profile ξW and floor profile ξF are re-defined for the local
coordinate system.
At a given position x, the contact stress profile σRmay
be calculated using the total interpenetration profile u0R at said position. This is done in a manner similar to that which was used in [9], where the force in a given contact spring is set to change with the square root of its deflec-tion in order to agree with Hertzian contact mechanics. In the discretized area between the wheel and the floor across which the calculation is performed, any points found to have negative stresses are assumed to be out of contact, and are thus set to zero. The stress profile may then be integrated across the entire contact area to obtain the full contact force due to roughness FR(x).
Once the contact force is known, the vertical wheel po-sition zW may be calculated using a convolution of the
(represented in Equation (7) by the symbol ∗). This results in an interdependency of Equations (3) and (5) to (8). To alleviate this, the vertical wheel position at x is used to calculate the total interpenetration profile at x + ∆x. For position x = 0, a vertical wheel position of zW(0) = 0 is
assumed.
Using this procedure, the wheel/floor interpenetration, contact force, and resulting wheel movement may be cal-culated for each position. At the end of the rolling event, the entire response history of the wheel and the floor is known.
2.3 Discrete Irregularities
Wheel flats are accounted for in the rolling noise model by modifying the wheel profile ξW at each discretized time
step. For a wheel of radius r, an ideal wheel flat may mod-eled as a chord of the wheel circumference, as shown in Figure 6. The flat depth hW, ideal flat length l0,W, and
ideal center angle Φ0are related by [19]
Φ0 2 = sin −1 l0,W 2r = cos−1 1 − hW r (12) The profile of such a wheel may be described by
R0,φ(φ) = ( r |φ| > Φ20 rcosΦ02 cos φ |φ| < Φ20 (13)
where φ is evaluated over the interval [−π, π].
Figure 6. Geometry of an ideal and rounded wheel flat. Alternatively, some rolling noise models have had suc-cess using a rounded wheel flat profile, which replaces the hard corner transition between flat and curved with one which has a more gentle profile [19–21]. This is intended to represent a wheel flat which has been in place for a pe-riod of time, and which has had its edges “worn down” from its ideal profile. Such a rounded profile may be rep-resented by [19] Φ 2 = sin −1 lW 2r = cos−1 1 −hW r (14) Rφ(φ) = (r |φ| > Φ2 R −hW 2 h 1 + cos2πφΦ i |φ| < Φ2 (15)
One may recall that the wheel radius rxis modified in
the contact model in order to keep agreement with Hertzian contact mechanics and account for the non-linear deflec-tion of the contact springs. Consequently, in order to map the flat spot to the wheel of reduced radius rx0, the flat
height is reduced here by the same amount.
h0W = hW
rx0 rx
(16)
Similarly, the reduction of the wheel radius implicitly means that the wheel will make a greater number of revo-lutions across the floor for a given rolling event than what occurs in reality. This will result in an increase in the num-ber of wheel flat impacts over a given rolling distance (in the case of a cylindrical wheel, the number of impacts will increase by a factor of two). This is corrected for by the following relation.
∆φ = ∆x rx
(17) where ∆x is the spatial resolution of the interpenetration profile in the global coordinate system, and ∆φ is the an-gular spacing of the wheel profile at a given moment in time t in polar coordinates.
At each discretized time step in the model, the formu-lation of ξW used in Equation (3) is updated as the wheel
flat moves through the vicinity of the contact area. A similar approach is taken to account for the presence of floor joints in the rolling model. As shown in Figure 7, this is added directly into the rolling model by defining the floor profile in the global coordinate system (x, z) as the joint depth −hF within ±lF/2 of the joint center, and
zero everywhere else. The joint centers are known based on the total rolling distance L and the length of an individ-ual floor tile. Just as the wheel flat height was multiplied by rx0/rx in order to correctly map the flat profile to the
reduced wheel radius, the same procedure is applied here, but this time to the floor joint length lF.
l0F = lF
r0 x
rx
(18)
Figure 7. Geometry of a simple floor joint.
2.4 Propagation Model
wheel response to a radiated sound power. This is done using [22] Πrad= 1 8π2 Z 2π 0 Z ∞ 0 Fflex(f ) Fref 2 Z0σfinite |ZsTMM(f ) + ZB,∞(f )| 2krdkrdφ ! (19)
where Fflexis the injected force, Fref is a reference force
of 1 N, Z0 is the acoustical impedance of air, ZsTMM is
the impedance of the multilayered floor, ZB,∞is the
radi-ation impedance, and σfiniteis the “finite size” radiation
ef-ficiency. The transfer matrix method (TMM) is used to cal-culate the impedance of the multilayered floor ZsTMM for
each wavenumber couple (kx, ky) or (kr, φ). In essence,
the radiated power of an infinite plate is calculated, then spatially windowed to achieve the effective radiation of a plate with size equal to the actual floor.
The injected flexural force Fflexdepends on the blocked
force, which is taken to be the contact force FR(f ). This
takes into account the impedance of both the wheel and floor, using |Fflex(f )| 2 = ZsTMM(f ) ZsTMM(f ) + Zexc(f ) 2 |FR(f )| 2 (20)
Where Zexc is the wheel impedance, estimated as the
blocked force divided by the time derivative of the verti-cal wheel position.
As with the wheel impedance, the floor impedance may theoretically be calculated by dividing the injected force by the floor velocity. However, as the injected force in cal-culated from the contact force found in the contact model, it may appear that it cannot be known ahead of time. To al-leviate this problem, the floor impedance is pre-calculated using an injected point force of 1 N. This essentially yields a unit floor impedance, which when used in Equation (20) in conjugation with the blocked force, will yield the correct injected flexural force.
3. ROUGHNESS MEASUREMENT
In order to provide an accurate estimation of the relative roughness between the wheel and the floor as input to the rolling noise model, roughness samples from two cylindri-cal wheels and three floors were measured. This was done using a Nikon LC15DX 3D scanner with an average spatial resolution of 22 µm.
The floors measured were polished concrete, a smooth PVC floor covering, and a rough PVC floor covering. The two PVC floors were identical in construction, the only difference being their surface roughness. A 3.5 cm wide section was measured along the full length of each floor sample: 20 cm for the concrete and 60 cm for the PVC.
The two wheels were identical in material composition and overall geometry (r = 5 cm, w = 3.5 cm), with the difference being that one contained a series six of non-periodic flat spots around its circumference (each flat spot
having a depth of 0.5 mm). The entire rolling surface was captured, giving a width equal to the wheel width, and length equal to the wheel’s circumference.
4. EXPERIMENTAL RESULTS
The model was compared to experimental results gathered from a rolling noise test performed at Level Acoustics and Vibration in Eindhoven, the Netherlands. The sound gen-erated by a rolling test trolley was captured in a two-story transmission room: with the trolley being rolled in the top (emission) room, and the sound measured in the bottom (reception) rooms. The measurement procedure outlined in ISO 10140-3 [1] for laboratory testing of a tapping ma-chine was used. The two rooms were separated by a 10 cm thick concrete floor, which was decoupled from the sur-rounding floor to eliminate sound transmitted to the recep-tion room via flanking transfer paths.
The test trolley used was a simple two-wheel design, as it allows the rolling noise itself to be accurately repre-sented while eliminating all other sources of noise which are beyond the scope of this model (e.g. rattling noise). The three floor types and two wheel types which had their roughness profiles measured were tested, for a total of six wheel/floor combinations.
4.1 Normalized sound pressure level
The model and experimental results were first compared using their normalized sound pressure level in the recep-tion room Ln, calculated from ISO 10140-3 [1]. This takes
into account the absorption of the reception room in calcu-lating the sound pressure level of the rolling trolley.
(a) Smooth wheel
(b) Ideal flat wheel
(c) Rounded flat wheel
Figure 8. Normalized sound pressure level in the reception room for varying floor covering: model versus experiment. For all curves, the trolley speed was 0.9 m/s and the added load was 10 kg.
—
model concrete floor,- - -experiment concrete floor,—
model, rough PVC floor,- - - experi-ment, rough PVC floor,—
model smooth PVC floor,-experiment, smooth PVC floor.
to 10 dB, so any values below this were corrected to this level. This can be seen particularly in the results for the smooth wheel above 1 000 Hz.
4.2 Relative normalized sound pressure level
An alternative way of visualizing the data is to calculate the reduction in normalized sound pressure level ∆Ln[1].
This is found by subtracting the Lnwith a floor covering in
place from the Lnof bare concrete. This is done not only
because the benefit of a floor covering is often the primary quantity of interest, but also because doing do helps re-move variations in experimental data, as well as modeling uncertainties, which may be present in the absolute level results.
Figure 9 shows the benefit of adding a floor covering, as computed with both model and experimental data. There is quite a bit less difference between the three wheel types in the ∆Ln plots than in the Ln plots, indicating that the
floors provide more or less a similar benefit regardless of wheel type. Additionally, while the model is not perfectly accurate at estimating the absolute sound level Ln, it is
much more accurate at estimating the reduction in sound level ∆Ln. A greater deviation occurs in the high
fre-quency range for the smooth wheel, where the model has a tendency to over-predict above 500 Hz. This is due to the weak excitation of the smooth wheels. The curves shown here were not corrected to the background noise level of 10 dB, so the behavior of the curves in the high frequency range are true to reality.
5. DISCUSSION
The primary objective of this work was to develop a rolling noise model that can capture the physical phenom-ena present in indoor rolling contact, as well as predict the sound level benefit of adding a floor covering to a given floor. The model which was developed has been specif-ically adapted for the characteristics of rolling in multi-story buildings, such as highly elastic wheels, multi-layer floors, and sound propagation via vertical structure-borne transfer paths. This is a domain which has hereto been left unexplored. As such, it serves as a ground-breaking look into the problem of indoor rolling noise. This work may be a platform to spur further exploration into the field of indoor rolling noise.
An important observation to make when comparing the two plots in Figure 8 is that, while the addition of a PVC floor covering aids in reducing the overall noise level for a given wheel, the flat wheel on PVC is still louder than the smooth wheel on bare concrete. This validates the ne-cessity for a model which is capable of accounting for dis-crete irregularities, as their presence can quickly eliminate any benefit that a quieter material (whether for the floor or wheel) would provide. The smooth wheel does not al-ways provide a sufficient signal to noise ratio for precise measurements, particularly in the presence of a soft floor covering.
(a) Smooth wheel
(b) Ideal flat wheel
(c) Rounded flat wheel
Figure 9. Benefit from applying the rough and smooth PVC floor coverings: model versus experiment. For all curves, the trolley speed was 0.9 m/s and the added load was 10 kg.
—
model concrete floor,- - -experiment con-crete floor,—
model, rough PVC floor,- - -experiment, rough PVC floor,—
model smooth PVC floor,- - - exper-iment, smooth PVC floor.noise, such as the tapping machine. However, because impact noise is starkly different than rolling noise, floor systems which have been developed for reducing impact noise will not necessarily exhibit high performance with rolling noise as well. This model may serve as a comple-ment to the existing indoor structure-borne noise modeling
solutions: aiding in the development of building materials which have better acoustic performance for a wider range of sound sources.
As is the case with any numerical model, the results are only as good as the inputs. In addition to the roughness profiles, experimental data regarding the elastic properties of the floor (in particular, the dynamic loss factor) are nec-essary for this model. It is known that the damping of the floor has a large effect on the transmitted structure-borne noise when being excited by a mechanical source [23]. While extensive measures were taken to obtain highly ac-curate roughness profiles, improved measurement of the wheel and floor material properties may provide greater accuracy in the overall model results.
The test trolley used was developed to produce a rolling sound which was as regular as possible: free of extraneous signatures such as rattling or other trolley vibrations. Just as there exists a standard tapping machine for normalizing the measurement process of impact noise across various test locations, the possibility of a standard rolling device also exists. The development of such a device could aid in the furthering the field of indoor rolling noise research, as it would allow a congruent means of comparing rolling noise measurements gathered in different scenarios. For further discussion on this topic, see [24].
6. CONCLUSION
This work presents an improved time-domain model for predicting rolling noise in buildings. The model is able to predict with relative accuracy the radiated sound power of a wheeled trolley rolling across a floor while accounting for the roughness of the floor & wheel, the elastic param-eters of the floor & wheel, the speed of the trolley, and the load of the trolley. The model is also able to account for discrete irregularities, such as wheel flat spots and floor joints. A model such as this one may be used as a supple-ment to the existing models for floor impact noise, in order to test different wheel/floor material combinations to see which ones result in the greatest reduction in noise. It may also be used to identify trends in order to guide the devel-opment of new vibro-acoustic materials which are effective at both reducing both impact noise and rolling noise.
7. ACKNOWLEDGMENTS
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